Symmetry properties of generalized graph truncations

In the generalized truncation construction, one replaces each vertex of a $k$-regular graph $\Gamma$ with a copy of a graph $\Upsilon$ of order $k$. We investigate the symmetry properties of the graphs constructed in this way, especially in connection to the symmetry properties of the graphs $\Gamma$ and $\Upsilon$ used in the construction. We demonstrate the usefulness of our results by using them to obtain a classification of cubic vertex-transitive graphs of girths $3$, $4$, and $5$.